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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4563, base_ring=CyclotomicField(78))
M = H._module
chi = DirichletCharacter(H, M([0,5]))
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pari:[g,chi] = znchar(Mod(3052,4563))
\(\chi_{4563}(82,\cdot)\)
\(\chi_{4563}(244,\cdot)\)
\(\chi_{4563}(433,\cdot)\)
\(\chi_{4563}(595,\cdot)\)
\(\chi_{4563}(784,\cdot)\)
\(\chi_{4563}(946,\cdot)\)
\(\chi_{4563}(1135,\cdot)\)
\(\chi_{4563}(1297,\cdot)\)
\(\chi_{4563}(1486,\cdot)\)
\(\chi_{4563}(1648,\cdot)\)
\(\chi_{4563}(1999,\cdot)\)
\(\chi_{4563}(2188,\cdot)\)
\(\chi_{4563}(2350,\cdot)\)
\(\chi_{4563}(2539,\cdot)\)
\(\chi_{4563}(2701,\cdot)\)
\(\chi_{4563}(2890,\cdot)\)
\(\chi_{4563}(3052,\cdot)\)
\(\chi_{4563}(3241,\cdot)\)
\(\chi_{4563}(3592,\cdot)\)
\(\chi_{4563}(3754,\cdot)\)
\(\chi_{4563}(3943,\cdot)\)
\(\chi_{4563}(4105,\cdot)\)
\(\chi_{4563}(4294,\cdot)\)
\(\chi_{4563}(4456,\cdot)\)
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sage:chi.galois_orbit()
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pari:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
\((677,3889)\) → \((1,e\left(\frac{5}{78}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 4563 }(3052, a) \) |
\(1\) | \(1\) | \(e\left(\frac{5}{78}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{67}{78}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{47}{78}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{14}{39}\right)\) |
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sage:chi.jacobi_sum(n)
